Shot noise is generated by electrical current fluctuations arising from the discrete nature of charge carriers. Shot noise can be used as an indicator of the correlation between charged carriers.1–41. L. S. Levitov and G. B. Lesovik, “ Charge distribution in quantum shot noise,” JETP Lett. 58, 230 (1993).2. Y. Blanter and M. Büttiker, “ Shot noise in mesoscopic conductors,” Phys. Rep. 336, 1–166 (2000). https://doi.org/10.1016/S0370-1573(99)00123-43. S. Ghosh, H. Surdi, F. Kargar, F. A. Koeck, S. Rumyantsev, S. Goodnick, R. J. Nemanich, and A. A. Balandin, “ Excess noise in high-current diamond diodes,” Appl. Phys. Lett. 120, 062103 (2022). https://doi.org/10.1063/5.00833834. D. Chevallier, T. Jonckheere, E. Paladino, G. Falci, and T. Martin, “ Detection of finite-frequency photoassisted shot noise with a resonant circuit,” Phys. Rev. B 81, 205411 (2010). https://doi.org/10.1103/PhysRevB.81.205411 A well-known characterization of shot noise is the Fano factor, F, which is defined as the ratio between the actual shot noise and the Poisson shot noise with F = 1. Prominent examples of systems with F≠1 include the super-Poissonian shot noise with F > 1 in zero-dimensional quantum dots5,65. E. Onac, F. Balestro, B. Trauzettel, C. F. Lodewijk, and L. P. Kouwenhoven, “ Shot-noise detection in a carbon nanotube quantum dot,” Phys. Rev. Lett. 96, 026803 (2006). https://doi.org/10.1103/PhysRevLett.96.0268036. M. C. Harabula, V. Ranjan, R. Haller, G. Fülöp, and C. Schönenberger, “ Blocking-state influence on shot noise and conductance in quantum dots,” Phys. Rev. B 97, 115403 (2018). https://doi.org/10.1103/PhysRevB.97.115403 and the sub-Poissonian shot noise with F = 1/3 in both disordered conductors7,87. K. Nagaev, “ On the shot noise in dirty metal contacts,” Phys. Lett. A 169, 103–107 (1992). https://doi.org/10.1016/0375-9601(92)90814-38. C. W. J. Beenakker and M. Büttiker, “ Suppression of shot noise in metallic diffusive conductors,” Phys. Rev. B 46, 1889(R) (1992). https://doi.org/10.1103/PhysRevB.46.1889 and graphene.9–119. J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker, “ Sub-Poissonian shot noise in graphene,” Phys. Rev. Lett. 96, 246802 (2006). https://doi.org/10.1103/PhysRevLett.96.24680210. R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, and P. J. Hakonen, “ Shot noise measurements in graphene,” Solid State Commun. 149, 1050–1055 (2009). https://doi.org/10.1016/j.ssc.2009.02.04611. R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, and P. J. Hakonen, “ Shot noise in ballistic graphene,” Phys. Rev. Lett. 100, 196802 (2008). https://doi.org/10.1103/PhysRevLett.100.196802 Furthermore, the well-celebrated maximal Fano factor value in graphene is shown to be associated with the minimal conductivity in orders of e2/ℏ.99. J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker, “ Sub-Poissonian shot noise in graphene,” Phys. Rev. Lett. 96, 246802 (2006). https://doi.org/10.1103/PhysRevLett.96.246802 Importantly, shot noise provides a useful tool to probe the quantum transport properties of an electronic system1212. C. Beenakker and C. Schönenberger, “ Quantum shot noise,” Phys. Today 56(5), 37–42 (2003). https://doi.org/10.1063/1.1583532 and has been widely employed in the experimental studies of graphene and their heterostructures.13,1413. M. R. Sahu, A. K. Paul, A. Soori, K. Watanabe, T. Taniguchi, S. Mukerjee, and A. Das, “ Enhanced shot noise at bilayer graphene–superconductor junction,” Phys. Rev. B 100, 235414 (2019). https://doi.org/10.1103/PhysRevB.100.23541414. N. Kumada, F. Parmentier, H. Hibino, D. Glattli, and P. Roulleau, “ Shot noise generated by graphene p-n junctions in the quantum hall effect regime,” Nat. Commun. 6, 8068 (2015). https://doi.org/10.1038/ncomms9068Two-dimensional (2D) semi-Dirac material (SDM) represents an interesting system that simultaneously host linear (relativistic) energy dispersion in one direction and parabolic (nonrelativistic) energy dispersion in the orthogonal direction.15–1715. S. Banerjee, R. R. Singh, V. Pardo, and W. E. Pickett, “ Tight-binding modeling and low-energy behavior of the semi-dirac point,” Phys. Rev. Lett. 103, 016402 (2009). https://doi.org/10.1103/PhysRevLett.103.01640216. S. Banerjee and W. E. Pickett, “ Phenomenology of a semi-Dirac semi-Weyl semimetal,” Phys. Rev. B 86, 075124 (2012). https://doi.org/10.1103/PhysRevB.86.07512417. H. Huang, Z. Liu, H. Zhang, W. Duan, and D. Vanderbilt, “ Emergence of a chern-insulating state from a semi-Dirac dispersion,” Phys. Rev. B 92, 161115(R) (2015). https://doi.org/10.1103/PhysRevB.92.161115 SDMs have been realized in a large variety of systems, including (TiO2)m/(VO2)n nanostructure,1818. V. Pardo and W. E. Pickett, “ Half-metallic semi-dirac-point generated by quantum confinement in TiO2/VO2 nanostructures,” Phys. Rev. Lett. 102, 166803 (2009). https://doi.org/10.1103/PhysRevLett.102.166803 strained organic salt,1919. S. Katayama, A. Kobayashi, and Y. Suzumura, “ Pressure-induced zero-gap semiconducting state in organic conductor α-(BEDT-TTF)2I3 salt,” J. Phys. Soc. Jpn. 75, 054705 (2006). https://doi.org/10.1143/JPSJ.75.054705 photonic crystals,2020. Y. Wu, “ A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22, 1906 (2014). https://doi.org/10.1364/OE.22.001906 Bi1−x Sbx,2121. S. Tang and M. S. Dresselhaus, “ Constructing a large variety of Dirac-cone materials in the Bi1-x Sbx thin film system,” Nanoscale 4, 7786–7790 (2012). https://doi.org/10.1039/c2nr32436a striped boron sheet,2222. H. Zhang, Y. Xie, Z. Zhang, C. Zhong, Y. Li, Z. Chen, and Y. Chen, “ Dirac nodal lines and tilted semi-Dirac cones coexisting in a striped boron sheet,” J. Phys. Chem. Lett. 8, 1707–1713 (2017). https://doi.org/10.1021/acs.jpclett.7b00452 on surface states of topological insulators,23,2423. Q. Li, P. Ghosh, J. D. Sau, S. Tewari, and S. Das Sarma, “ Anisotropic surface transport in topological insulators in proximity to a helical spin density wave,” Phys. Rev. B 83, 085110 (2011). https://doi.org/10.1103/PhysRevB.83.08511024. F. Zhai, P. Mu, and K. Chang, “ Energy spectrum of Dirac electrons on the surface of a topological insulator modulated by a spiral magnetization superlattice,” Phys. Rev. B 83, 195402 (2011). https://doi.org/10.1103/PhysRevB.83.195402 or in non-centrosymmetric systems,2525. S. Park and B.-J. Yang, “ Classification of accidental band crossings and emergent semimetals in two-dimensional noncentrosymmetric systems,” Phys. Rev. B 96, 125127 (2017). https://doi.org/10.1103/PhysRevB.96.125127 such as phosphorus-based materials,26–3226. S. S. Baik, K. S. Kim, Y. Yi, and H. J. Choi, “ Emergence of two-dimensional massless Dirac fermions, chiral pseudospins, and Berrys phase in potassium doped few-layer black phosphorus,” Nano Lett. 15, 7788–7793 (2015). https://doi.org/10.1021/acs.nanolett.5b0410627. B. Ghosh, B. Singh, R. Prasad, and A. Agarwal, “ Electric-field tunable Dirac semimetal state in phosphorene thin films,” Phys. Rev. B 94, 205426 (2016). https://doi.org/10.1103/PhysRevB.94.20542628. Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, “ Switching a normal insulator into a topological insulator via electric field with application to phosphorene,” Nano Lett. 15, 1222–1228 (2015). https://doi.org/10.1021/nl504376929. J. Kim, S. S. Baik, S. W. Jung, Y. Sohn, S. H. Ryu, H. J. Choi, B.-J. Yang, and K. S. Kim, “ Two-dimensional Dirac fermions protected by space-time inversion symmetry in black phosphorus,” Phys. Rev. Lett. 119, 226801 (2017). https://doi.org/10.1103/PhysRevLett.119.22680130. S. Adhikary, S. Mohakud, and S. Dutta, “ Engineering anisotropic Klein tunneling in black phosphorene through elemental substitution,” Phys. Status Solidi Basic Res. 258, 2100071 (2021). https://doi.org/10.1002/pssb.20210007131. C. Wang, Q. Xia, Y. Nie, and G. Guo, “ Strain-induced gap transition and anisotropic Dirac-like cones in monolayer and bilayer phosphorene,” J. Appl. Phys. 117, 124302 (2015). https://doi.org/10.1063/1.491625432. M. Yarmohammadi, M. Mortezaei, and K. Mirabbaszadeh, “ Anisotropic basic electronic properties of strained black phosphorene,” Phys. E Low-dimensional Syst. Nanostruct. 124, 114323 (2020). https://doi.org/10.1016/j.physe.2020.114323 monolayer arsenene,3333. C. Wang, Q. Xia, Y. Nie, M. Rahman, and G. Guo, “ Strain engineering band gap, effective mass and anisotropic Dirac-like cone in monolayer arsenene,” AIP Adv. 6, 035204 (2016). https://doi.org/10.1063/1.4943548 silicene oxide,3434. C. Zhong, Y. Chen, Y. Xie, Y. Y. Sun, and S. Zhang, “ Semi-Dirac semimetal in silicene oxide,” Phys. Chem. Chem. Phys. 19, 3820–3825 (2017). https://doi.org/10.1039/C6CP08439G and polariton lattice,3535. B. Real, O. Jamadi, M. Milićević, N. Pernet, P. St-Jean, T. Ozawa, G. Montambaux, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, S. Ravets, J. Bloch, and A. Amo, “ Semi-Dirac transport and anisotropic localization in polariton honeycomb lattices,” Phys. Rev. Lett. 125, 186601 (2020). https://doi.org/10.1103/PhysRevLett.125.186601 and also in α-dice lattice36–3836. J. P. Carbotte, K. R. Bryenton, and E. J. Nicol, “ Optical properties of a semi-Dirac material,” Phys. Rev. B 99, 115406 (2019). https://doi.org/10.1103/PhysRevB.99.11540637. L. Mandhour and F. Bouhadida, “ Klein tunneling in deformed α−T3 lattice,” arXiv:2004.10144 (2020).38. E. Illes and E. J. Nicol, “ Klein tunneling in the α-T3 model,” Phys. Rev. B 95, 235432 (2017). https://doi.org/10.1103/PhysRevB.95.235432 with higher pseudospin. The electronic transport and shot noise of SDM have been studied extensively in previous works, which establish a Fano factor of F = 1 and F≈0.179 at the band insulator phase with nonzero bandgap and at the semimetallic gapless limit, respectively.38–4738. E. Illes and E. J. Nicol, “ Klein tunneling in the α-T3 model,” Phys. Rev. B 95, 235432 (2017). https://doi.org/10.1103/PhysRevB.95.23543239. K. Ghasemian, M. R. Setare, D. Jahani, and J. Naji, “ Klein tunneling of semi-Dirac-like fermions in graphene,” Europhys. Lett. 136, 17005 (2021). https://doi.org/10.1209/0295-5075/ac336140. F. Zhai and J. Wang, “ Shot noise in systems with semi-Dirac points,” J. Appl. Phys. 116, 063704 (2014). https://doi.org/10.1063/1.489284341. Y. Betancur-Ocampo, F. Leyvraz, and T. Stegmann, “ Electron optics in phosphorene pn junctions: Negative reflection and anti-super-Klein tunneling,” Nano Lett. 19, 7760–7769 (2019). https://doi.org/10.1021/acs.nanolett.9b0272042. Y. W. Choi and H. J. Choi, “ Anisotropic pseudospin tunneling in two-dimensional black phosphorus junctions,” 2D Mater. 8, 035024 (2021). https://doi.org/10.1088/2053-1583/abf81043. Z. Li, T. Cao, M. Wu, and S. G. Louie, “ Generation of anisotropic massless Dirac fermions and asymmetric Klein tunneling in few-layer black phosphorus superlattices,” Nano Lett. 17, 2280–2286 (2017). https://doi.org/10.1021/acs.nanolett.6b0494244. S. W. Jung, S. H. Ryu, W. J. Shin, Y. Sohn, M. Huh, R. J. Koch, C. Jozwiak, E. Rotenberg, A. Bostwick, and K. S. Kim, “ Black phosphorus as a bipolar pseudospin semiconductor,” Nat. Mater. 19, 277–281 (2020). https://doi.org/10.1038/s41563-019-0590-245. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.24541046. K. Saha, R. Nandkishore, and S. A. Parameswaran, “ Valley-selective Landau-Zener oscillations in semi-Dirac p-n junctions,” Phys. Rev. B 96, 045424 (2017). https://doi.org/10.1103/PhysRevB.96.04542447. S. Rostamzadeh and M. Sarisaman, “ Charge-pseudospin coupled diffusion in semi-Dirac graphene: Pseudospin assisted valley transport,” New J. Phys. 24, 083026 (2022). https://doi.org/10.1088/1367-2630/ac86e8 Nevertheless, 2D SDM can undergo complex topological phase transitions. Beyond the band insulating and the semimetallic regime, SDM can exhibit a band inversion phase2525. S. Park and B.-J. Yang, “ Classification of accidental band crossings and emergent semimetals in two-dimensional noncentrosymmetric systems,” Phys. Rev. B 96, 125127 (2017). https://doi.org/10.1103/PhysRevB.96.125127 in which two distinct Dirac cones emerge, thus rendering the band-inverted SDM a strong potential for valleytronic device applications.4545. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.245410 Nevertheless, the shot noise and conductance signatures of band-inverted SDM remain an open question thus far.In this work, we study the quantum transport of SDM near the topological phase transitions. Focusing on the quantum transport occurring along the relativistic direction—not covered in the previous quantum transport study,4545. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.245410 we observe the intriguing coexistence of massless9,489. J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker, “ Sub-Poissonian shot noise in graphene,” Phys. Rev. Lett. 96, 246802 (2006). https://doi.org/10.1103/PhysRevLett.96.24680248. M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “ Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006). https://doi.org/10.1038/nphys384 and massive Dirac fermions in the tunneling spectra at fixed transport channel at all quasiparticle energies, which is distinctive from that of bilayer graphene4949. W.-Y. He, Z.-D. Chu, and L. He, “ Chiral tunneling in a twisted graphene bilayer,” Phys. Rev. Lett. 111, 066803 (2013). https://doi.org/10.1103/PhysRevLett.111.066803 in which the massive and massless Dirac quasiparticles occur at various transversal momenta at different quasiparticle energies. We further calculate the conductance and Fano factors of 2D SDMs as the band topology changes continuously from band insulating to band inversion phases. Remarkably, we found that the Fano factor converges to sub-Poissonian shot noise with F≈1/3 and to Poissonian shot noise with F≈1 in the band inversion and insulating phase, respectively [Fig. 1(a)]. Such shot noise signatures have negligible thermal noise contributions, even at room temperature. Our findings reveal the exotic quantum transport behavior and the shot noise signatures of 2D SDM at various phases, thus uncovering shot noise as a useful tool in probing the band topology of 2D SDM.2D SDM can be described by a two-band effective Hamiltonian,50,5150. G. Montambaux, F. Piéchon, J.-N. Fuchs, and M. O. Goerbig, “ A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal,” Eur. Phys. J. B 72, 509–520 (2009). https://doi.org/10.1140/epjb/e2009-00383-051. G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “ Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80, 153412 (2009). https://doi.org/10.1103/PhysRevB.80.153412 Ĥ=(αkx2+Δ)σx+βkyσy,(1)with α=ℏ2/(2m*) and β=ℏvy, where m* and vy are the effective mass along x̂ and Fermi velocity along ŷ, respectively. A phase transition parameter, Δ, acts as a perturbation factor that continuously changes the band topology from the band insulating phase (Δ>0) and the semi-Dirac phase (Δ = 0) to the band inversion phase (Δ<0). We can nondimensionlize Eq. (1) by defining the characteristic momentum and energy (ℏk0=2m*vy and ε0=ℏk0vy) to obtain Ĥ=(kx2+Δ)σx+kyσy, which has the following energy dispersion: where ± indicates conduction/valence (+/–) bands. This parameter has been utilized for directional dependent transport40,52–5640. F. Zhai and J. Wang, “ Shot noise in systems with semi-Dirac points,” J. Appl. Phys. 116, 063704 (2014). https://doi.org/10.1063/1.489284352. A. Mawrie and B. Muralidharan, “ Direction-dependent giant optical conductivity in two-dimensional semi-Dirac materials,” Phys. Rev. B 99, 075415 (2019). https://doi.org/10.1103/PhysRevB.99.07541553. X. Zhou, W. Chen, and X. Zhu, “ Anisotropic magneto-optical absorption and linear dichroism in two-dimensional semi-Dirac electron systems,” Phys. Rev. B 104, 235403 (2021). https://doi.org/10.1103/PhysRevB.104.23540354. H. Y. Zhang, Y. M. Xiao, Q. N. Li, L. Ding, B. van Duppen, W. Xu, and F. M. Peeters, “ Anisotropic and tunable optical conductivity of a two-dimensional semi-Dirac system in the presence of elliptically polarized radiation,” Phys. Rev. B 105, 115423 (2022). https://doi.org/10.1103/PhysRevB.105.11542355. J. Kim, S. S. Baik, S. H. Ryu, Y. Sohn, S. Park, B.-G. Park, J. Denlinger, Y. Yi, H. J. Choi, and K. S. Kim, “ Observation of tunable band gap and anisotropic Dirac semimetal state in black phosphorus,” Science 349, 723–726 (2015). https://doi.org/10.1126/science.aaa648656. S. M. Cunha, D. R. da Costa, J. M. Pereira, R. N. C. Filho, B. V. Duppen, and F. M. Peeters, “ Tunneling properties in α-T3 lattices: Effects of symmetry-breaking terms,” Phys. Rev. B 105, 165402 (2022). https://doi.org/10.1103/PhysRevB.105.165402 and phase-dependent transport.36,44,57,5836. J. P. Carbotte, K. R. Bryenton, and E. J. Nicol, “ Optical properties of a semi-Dirac material,” Phys. Rev. B 99, 115406 (2019). https://doi.org/10.1103/PhysRevB.99.11540644. S. W. Jung, S. H. Ryu, W. J. Shin, Y. Sohn, M. Huh, R. J. Koch, C. Jozwiak, E. Rotenberg, A. Bostwick, and K. S. Kim, “ Black phosphorus as a bipolar pseudospin semiconductor,” Nat. Mater. 19, 277–281 (2020). https://doi.org/10.1038/s41563-019-0590-257. P. V. Sriluckshmy, K. Saha, and R. Moessner, “ Interplay between topology and disorder in a two-dimensional semi-Dirac material,” Phys. Rev. B 97, 024204 (2018). https://doi.org/10.1103/PhysRevB.97.02420458. K. Saha, “ Photoinduced Chern insulating states in semi-Dirac materials,” Phys. Rev. B 94, 081103(R) (2016). https://doi.org/10.1103/PhysRevB.94.081103 Owing to the presence of two inequivalent valleys in the band inversion phase (Δ<0), band-inverted 2D SDMs have been widely studied for potential applications in valleytronics.44–4744. S. W. Jung, S. H. Ryu, W. J. Shin, Y. Sohn, M. Huh, R. J. Koch, C. Jozwiak, E. Rotenberg, A. Bostwick, and K. S. Kim, “ Black phosphorus as a bipolar pseudospin semiconductor,” Nat. Mater. 19, 277–281 (2020). https://doi.org/10.1038/s41563-019-0590-245. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.24541046. K. Saha, R. Nandkishore, and S. A. Parameswaran, “ Valley-selective Landau-Zener oscillations in semi-Dirac p-n junctions,” Phys. Rev. B 96, 045424 (2017). https://doi.org/10.1103/PhysRevB.96.04542447. S. Rostamzadeh and M. Sarisaman, “ Charge-pseudospin coupled diffusion in semi-Dirac graphene: Pseudospin assisted valley transport,” New J. Phys. 24, 083026 (2022). https://doi.org/10.1088/1367-2630/ac86e8From Fig. 1(a), the SDM (Δ = 0) with a semi-Dirac point at (kx,ky)=(0,0) can become a band insulator (Δ>0) or exhibit a band inversion (Δ<0) phase with band crossings at (kx,ky)=(±kD,0), with kD=|Δ|.15,17,2515. S. Banerjee, R. R. Singh, V. Pardo, and W. E. Pickett, “ Tight-binding modeling and low-energy behavior of the semi-dirac point,” Phys. Rev. Lett. 103, 016402 (2009). https://doi.org/10.1103/PhysRevLett.103.01640217. H. Huang, Z. Liu, H. Zhang, W. Duan, and D. Vanderbilt, “ Emergence of a chern-insulating state from a semi-Dirac dispersion,” Phys. Rev. B 92, 161115(R) (2015). https://doi.org/10.1103/PhysRevB.92.16111525. S. Park and B.-J. Yang, “ Classification of accidental band crossings and emergent semimetals in two-dimensional noncentrosymmetric systems,” Phys. Rev. B 96, 125127 (2017). https://doi.org/10.1103/PhysRevB.96.125127 Interestingly, Eq. (1) in the band inversion regime can be decomposed into either the 1D massless and massive Dirac Hamiltonian [Fig. 1(c)] along the ky direction at kx=±kD and at kx = 0, respectively, Ĥ(Δ<0)={kyσy,kx=±kD;Δσx+kyσy,kx=0.(3)The quantum transport along the ky direction is, thus, expected to exhibit a mixture of massive and massive Dirac quasiparticles dictated in Eq. (3).The transmission probability T can be obtained by considering a scattering potential along ky [Fig. 1(b)] with U(y)=U0(Θ(y)−Θ(y−d)), where U0≡U0/ε0 and d≡d0k0 are the dimensionless potential height and barrier width, respectively. In the Lx≫d limit, the Hamiltonian decouples into a 1D eigenvalue equation along ky→−i∂y, with Ψj=Aj(εjkx2+Δ+ikj)eikjy+Bj(εjkx2+Δ−ikj)e−ikjy,(4)where the index j denotes the L (left), B (barrier), and (R) right regions. The energy and wavevector with index j denote εL/R=εk, εB=εk−U0≡εq with kL/R=ky=λεk2−(kx2+Δ)2, kB=qy=λ′εq2−(kx2+Δ)2, λ=sgn(εk) and λ′=sgn(εq). For the left incident wavefunction, AL = 1 and BR = 0 are enforced with the transmission coefficient t = AR. For only forward-moving electronic states in the region R, the wavevectors are enforced in Eq. (4) through ky>0 and qy<0. For the n–p–n junction, the conservation of current (non-dimensionlized), Ĵi+Ĵr=Ĵt with Ĵy=ℏ2Ψ†σ̂yΨ=ℏ2Ψ†σ̂yΨ, gives us the relation |r|2+|t|2=1. By matching the boundary conditions at y = 0 and d, we obtain T=|t|2 as T=4εk2εq2ky2qy24εk2εq2ky2qy2 cos2(λ′qyd)+sin2(λ′qyd)[(kx2+Δ)2U02+εk2qy2+εq2ky2]2,(5)in agreement with a band insulator for Δ>0 [top left panel of Fig. 2(a)] and a SDM4040. F. Zhai and J. Wang, “ Shot noise in systems with semi-Dirac points,” J. Appl. Phys. 116, 063704 (2014). https://doi.org/10.1063/1.4892843 for Δ = 0 by noticing that tan ϕ=(kx2+Δ)/ky and tan θ=(kx2+Δ)/qy [top right panel of Fig. 2(a)]. Remarkably, the rotational invariant of Eq. (2) around ẑ, i.e., [H,Rz]=0 with Rz being a rotation operator about the ẑ, allows Eq. (5) to fully capture the Klein tunneling behavior, as evident under rotation and incident angle.39,4339. K. Ghasemian, M. R. Setare, D. Jahani, and J. Naji, “ Klein tunneling of semi-Dirac-like fermions in graphene,” Europhys. Lett. 136, 17005 (2021). https://doi.org/10.1209/0295-5075/ac336143. Z. Li, T. Cao, M. Wu, and S. G. Louie, “ Generation of anisotropic massless Dirac fermions and asymmetric Klein tunneling in few-layer black phosphorus superlattices,” Nano Lett. 17, 2280–2286 (2017). https://doi.org/10.1021/acs.nanolett.6b04942 However, this cannot be generalized along the parabolic direction due to intervalley scattering.39–41,4539. K. Ghasemian, M. R. Setare, D. Jahani, and J. Naji, “ Klein tunneling of semi-Dirac-like fermions in graphene,” Europhys. Lett. 136, 17005 (2021). https://doi.org/10.1209/0295-5075/ac336140. F. Zhai and J. Wang, “ Shot noise in systems with semi-Dirac points,” J. Appl. Phys. 116, 063704 (2014). https://doi.org/10.1063/1.489284341. Y. Betancur-Ocampo, F. Leyvraz, and T. Stegmann, “ Electron optics in phosphorene pn junctions: Negative reflection and anti-super-Klein tunneling,” Nano Lett. 19, 7760–7769 (2019). https://doi.org/10.1021/acs.nanolett.9b0272045. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.245410We now focus on the band inversion phase for Δ<0 [bottom panel of Fig. 2(a)]. The tunneling behavior3737. L. Mandhour and F. Bouhadida, “ Klein tunneling in deformed α−T3 lattice,” arXiv:2004.10144 (2020). in Eq. (5) at different transverse momenta, kx, can be understood from the pseudospin texture4545. Y. S. Ang, S. A. Yang, C. Zhang, Z. Ma, and L. K. Ang, “ Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate,” Phys. Rev. B 96, 245410 (2017). https://doi.org/10.1103/PhysRevB.96.245410 (non-dimensionalized) shown in